A particle with charge e and mass m, moving along the X-axis with a uniform speed u enters a region where a uniform electric field E is acting along the Y-axis. The particle starts to move in a parabola. Its focal length (neglecting any effect of gravity) is

Options

A. `(2m u^(2))/(eE)`
B. `(eE)/(2m u^(2))`
C. `(m u)/(2eE)`
D. `(m u^(2))/(2 eE)`

Explanation

If the particle moves thrugh the electric field for a time t.
`t=(x)/(u)[x`= displacement along x=-axix]
`:.` Displacement along Y-axis,
`y=(1)/(2) a t^(2)=(1)/(2) ((eE)/(m))*(x^(2))/(u^(2))` [ `:. a`= acceleration of the particle `=(eE)/(m)]`
`:. x^(2)=(2m u^(2))/(eE)=y`
Now, comparing with the general equation of parabola,
`x^(2)=4'y`
Focal length of the parabola `a'=(2 m u^(2))/( 4eE)=( m u^(2))/( 2 eE)`
The option (D) is correct.